We review the recently found large-scale anomalies in the maps of temperature
anisotropies in the cosmic microwave background. These include alignments of
the largest modes of CMB anisotropy with each other and with geometry and
direction of motion of the Solar System, and the unusually low power at these
largest scales. We discuss these findings in relation to expectation from
standard inflationary cosmology, their statistical significance, the tools to
study them, and the various attempts to explain them.

This is a very clear review of the CMB large angle anomalies. In particular, the authors reply to the WMAP7 paper 1001.4758 claiming that there are no anomalies. The WMAP7 paper quoted the work 0909.2495 by Francis and Peacock arguing that the low quadrupole power and quadrupole-octopole alignment is due to the ISW effect of local structures and the paper 0911.5399 by Efstathiou, Ma and Hanson arguing that the large-angle correlation function is not anomalous.

The authors argue that if the local ISW explanation of Francis and Peacock is correct, this implies two new anomalies, namely an unusually low amplitude for the Sachs-Wolfe effect so that the ISW effect can dominate, and accidental cancellation between the SW and ISW anisotropy patterns.

Efstathiou, Ma and Hanson claimed to determine the large-angle correlationsfrom the cut sky without assuming statistical isotropy. The present authors argue that the method implicitly assumes statistical isotropy by way of independence of low and high multipoles, and note that their method which works directly in angular (as opposed to multipole) space does not involve such an assumption and is therefore more robust. (In fact, I cannot understand how one could possibly reconstruct large-angle statistics from a sky where 25% has been cut without making assumptions about about what is missing.)

Yup, I think that I basically agree with that. Using QML estimates for the CMB power spectrum on the cut-sky at low-l can be viewed as a sort of statement about the independence of low and high multipoles-- though it's probably better to think of it as a statement that the CMB power spectrum at low-L is sufficiently red that the reconstruction error induced by the 'noise' generated by higher multipoles is negligible. This is what the comparison of several QML Cl estimates with different lmax is testing (0911.5399). Of course, if the signal inside the cut were completely different from that outside this would be wrong to some extent. Suppose that inside the cut you had an imprint of Stephen Colbert's face, for example, then you would want to use a less conservative mask. But there is no imprint of Colbert's face-- the original motivation for using a cut sky rather than e.g. the ILC map was that the level of residual foreground contamination inside the cut was not well understood (1001.4758). Now it is, so you have a problem-- why does the cut-sky pseudo-Cl based correlation function seem anomalous, while the ILC reconstruction does not? Is there something funny (some might argue that a Colbert image would not be funny) in the galactic plane which restores the large angle correlations? No-- the ILC map looks exactly as you would expect from the cut-sky and the redness of the CMB (keep in mind, if this weren't the case it would represent another puzzle of the type which you've mentioned potentially troubles the Francis & Peacock arguments -- if you want to say that the data outside the cut is anomalous, but that inside is not, why are they correlated such that the pair looks perfectly reasonable?). I agree that the pseudo-Cl estimate of the correlation function would be the more robust choice if you had a model in which the data inside the sky cut could contain something arbitrary, but if foregrounds have been accurately treated then we don't have such a model, and so the only reason you would use a pseudo-Cl correlation estimator is if you used it once and it gave you something 'interesting'-- i.e. a priori reasons.

Hiranya Peiris and I had a paper on this a couple of weeks ago, 1004.2706. Although it's a side issue for us, our paper does explain exactly why standard QML reconstructs the full sky Cl's with smaller errors than PCL, even for arbitrarily anisotropic theories (see Sec II and Appendix A, in particular figs 5 & 6). The significance of PCL here is that it is precisely equivalent to the pixel-based estimators that the Copi group use (see our Appendix B).

If you have something arbitrary in the cut which is not band-limited in harmonic space, then obviously no method will reconstruct it, neither PCL nor QML. (That's the whole aim of masking, to destroy unwanted, localized contamination.) The signature of that kind of contamination would be for QML and PCL reconstructions to both disagree with the full-sky analysis (see top left panel of our Figure 3). This is not what we see in the WMAP data for sensible sky-cuts.

All in all there is no doubt that QML is a more robust way than PCL or pixel techniques to estimate the full-sky Cl's or C(theta) from the cut sky.

However, one might still worry 'why' PCL techniques give such an unlikely result, regarding them not as an estimator for the full sky, but as a distinct quantity in their own right. You can trace that to (a) the low quadrupole and (b) the alignment of power on the full sky with the mask -- the power gets hidden behind the mask (our Section III). Incidentally the recent letter 1004.3784 does not see this, apparently because they only align the quadrupole and octupole; it's further necessary to constrain C_2 to be low, and to make the octupole power planar before aligning it approximately with the galactic plane. There are then further, more subtle alignments at l=5, 7 which give the effect its strong frequentist significance (see our Section III.B and Figure 1, left panels).

In terms of the ISW argument, the point is that any physical explanation of the low multipole anomalies must correlate the primary and ISW contributions to the final signal. That's very hard to realize in a physical theory. It doesn't make the signal 'more anomalous'; rather it makes the prior on any theory purporting to explain it very small -- i.e. it increases the barrier to claiming there is physical significance in the low-multipole anomalies. Combined with the very small likelihood gains available (our Section IV) this kind of argument shows the cut-sky angular correlation function does not constitute strong evidence against vanilla-\Lambda CDM.

Syksy Rasanen wrote:
I don't follow you. Why does any possible explanation have to correlate the primary and ISW anisotropies?

If there is missing power in some part of the sky on large angular scales, either the ISW isn't there at all or it is acting to cancel out the primary signal. If power comes from both primary and ISW anisotropies, the expected power on the summed sky is larger than that from the primary contribution alone -- unless the fields are correlated.

Syksy Rasanen wrote:I understand. But how does this establish that any possible explanation would have to correlate the primary and ISW anisotropies?

As you note in your top post, Francis and Peacock's results imply that the ISW and SW need to "accidentally" cancel in order for our z=0 CMB to have the observed alignments. George Efstathiou is saying a physical explanation needs to turn "accidentally" into a causal mechanism for this cancellation.

Having said that, if the ISW signal is somehow absent or much smaller amplitude than expected in the concordance model, then you don't need to correlate the signals, because in this case the signal with the alignments is coming from just the primary CMB. Or you can reverse this situation and have only ISW, and no SW on the large scales.

It doesn't really matter too much what one's view on these three possibilities are; the real point is that you can understand the cut sky C(theta) frequentist anomaly as arising from low l=2,3,5,7 alignments on the full sky, coupled with the low quadrupole.

Whatever the priors the ISW argument gives you, we've confirmed in 1004.2706 that no anisotropic Gaussian theories can give you a significantly better likelihood than Lambda-CDM does for the observed frequentist anomaly in the cut sky C(theta).

Andrew Pontzen wrote:Having said that, if the ISW signal is somehow absent or much smaller amplitude than expected in the concordance model, then you don't need to correlate the signals, because in this case the signal with the alignments is coming from just the primary CMB. Or you can reverse this situation and have only ISW, and no SW on the large scales.

OK, we agree that it is not the case that any possible physicsl explanation would require correlating the primary anisotropy and the ISW signal.

I agree that 1004.5602 is a very clear review of the issue, and
gives clear responses to several recent papers.

Andrew Pontzen wrote:...we've confirmed in 1004.2706 that no
anisotropic Gaussian theories can give you a significantly better
likelihood than Lambda-CDM does for the observed frequentist anomaly
in the cut sky C(theta).

It seems to me that 1004.2706 makes the assumption that the [tex]a_{lm}[/tex]'s
are drawn from (statistically) independent Gaussian distributions. Is
this correct? If yes, then 1004.2706 does not deal with one of the
main cosmological properties of exact-FLRW models discussed in1004.5602 (and 1004.5602 also considers going beyond exact-FLRW).

A physical model of the Universe in the sense of exact-FLRW models has
a constant-curvature 3-manifold as its comoving spatial section.
Friedmann, Lemaitre and Robertson (among others) pointed this out,
although the mathematics of constant-curvature 3-manifolds was still
poorly known at the time (see e.g. gr-qc/9605010,astro-ph/9901364). Copi et al. 1004.5602 refer to this quite clearly
in their 4th and 5th paragraphs, and elsewhere in the text. In the
3rd paragraph of Section V.E, Copi et al. point out that in an
exact-FLRW model, the distributions of the [tex]a_{lm}[/tex]'s are not, in
general, statistically independent distributions on the largest
scales. This is not just an issue of global versus local isotropy or
global versus local homogeneity. If you're looking at the same point
in comoving space when you're observing widely separated sky
positions, then those widely separated sky positions should be correlated
according to the correlation expected for closely separated points in
comoving space. One of Copi et al's major points is to consider the
possibility that the [tex]a_{lm}'s[/tex] may not be drawn from statistically
independent distributions, as well as to avoid other assumptions about
the fluctuation statistics if possible, since otherwise, circular
reasoning is used (Section 1.2.1 (b), astro-ph/9910272).

It is quite possible (but not necessary) for exact-FLRW models plus
density perturbations to be modelled as "anisotropic Gaussian",
i.e. the space is locally isotropic and globally anisotropic, and the
distributions of the amplitudes of the eigenmodes of the comoving
space can be drawn from Gaussian distributions. However, this
does not lead to full Gaussianity of the [tex]a_{lm}[/tex]'s for the lowest
l's. Independently, it is also not obvious that the amplitude
distributions of the comoving space eigenmodes should all be Gaussian
at the largest scales.

In any case, if by "anisotropic Gaussian theories" you only include
models in which the [tex]a_{lm}[/tex]'s are drawn from (statistically)
independent Gaussian distributions, then you have dealt with a
different question than one of the main questions raised by Copi et
al. in 1004.5602. [tex]C_{\theta}[/tex] can be calculated without requiring
the assumption that space is simply-connected. (Incidentally, "simply"
refers only to a topological property, not to the space itself. An
infinite flat 3D space as a physical model is not simple.)

Boud Roukema wrote:
It seems to me that 1004.2706 makes the assumption that the [tex]a_{lm}[/tex]'s
are drawn from (statistically) independent Gaussian distributions. Is
this correct? If yes, then 1004.2706 does not deal with one of the
main cosmological properties of exact-FLRW models discussed in1004.5602 (and 1004.5602 also considers going beyond exact-FLRW).

No, that's incorrect. The 'designer' theory in 1004.2706 allows complete freedom in the covariance matrix, meaning the a_lm's can be correlated in any way whatsoever. The restrictions are just that the theory is Gaussian and has zero mean.

Boud Roukema wrote:
It seems to me that 1004.2706 makes the assumption that the [tex]a_{lm}[/tex]'s
are drawn from (statistically) independent Gaussian distributions. Is
this correct? If yes, then 1004.2706 does not deal with one of the
main cosmological properties of exact-FLRW models discussed in1004.5602 (and 1004.5602 also considers going beyond exact-FLRW).

No, that's incorrect. The 'designer' theory in 1004.2706 allows complete freedom in the covariance matrix, meaning the a_lm's can be correlated in any way whatsoever. The restrictions are just that the theory is Gaussian and has zero mean.

And the obvious requirement that the covariance matrix of the theory is positive definite.

Boud Roukema wrote:I agree that 1004.5602 is a very clear review of the issue, and gives clear responses to several recent papers.

I didn't see that it had responded to any of the points made in 1004.2706. We would obviously welcome a discussion of these points here, if the authors are watching.

The point is that the angular correlation is unusually low on the only part of the sky that can be reliably observed. All of the talk about theory expectation muddles this simple fact I think.

For example, Andrew P writes that "you can understand the cut sky C(theta) frequentist anomaly as arising from low l=2,3,5,7 alignments on the full sky, coupled with the low quadrupole". That sounds awfully complicated. While this bizarre set of alignments might indeed the true reason behind the low cut-sky C(theta), I don't see how it, by itself, alleviates the anomaly.

The reason for low cut-sky C(theta) also does not appear to be fortuitous alignment of the cut and the large-scale power, as found in Sarkar et al 1004.3784 that was mentioned above. The 'aligned' Monte Carlo sky realizations in this paper, in addition to the Q-O alignment, did not enforce the low C_2 (as Pontzen noticed), but did effectively enforce the planar octopole via the statistics S^(3,3) in that paper. (A fair comparison would look for the covariance between all-alignment vs. all-power constrained realizations, which is what Sarkar et al did, and not some mix). So it doesn't appear that the alignments-imply-missing-correlations type of explanation is at work.

As for the ISW, I don't follow. In our review paper, we basically agree with Andrew's comments how low large-angle power would imply fortuitous cancellation between SW and ISW signals, and therefore "mak[ing] the prior on any theory purporting to explain it very small". But how is this an argument against the observed low correlations?

Dragan Huterer wrote:The point is that the angular correlation is unusually low on the only part of the sky that can be reliably observed. All of the talk about theory expectation muddles this simple fact I think.

We obviously start from very different perspectives. As far as I can tell, in the absence of theory, it's impossible to quantify unambiguously the significance of any anomaly. This follows from basic Bayesian considerations. If we can't agree on even that, I don't think there's any hope.

Dragan Huterer wrote:
For example, Andrew P writes that "you can understand the cut sky C(theta) frequentist anomaly as arising from low l=2,3,5,7 alignments on the full sky, coupled with the low quadrupole". That sounds awfully complicated. While this bizarre set of alignments might indeed the true reason behind the low cut-sky C(theta), I don't see how it, by itself, alleviates the anomaly.

No-one ever claimed it did -- as we say explicitly in our paper: "[this] does not by itself determine whether the anomaly might point to theories beyond the concordance model". It's a tool to allow us to construct simple theories which mimic the C(theta)^cut behaviour.

Dragan Huterer wrote:
The reason for low cut-sky C(theta) also does not appear to be fortuitous alignment of the cut and the large-scale power, as found in Sarkar et al 1004.3784 that was mentioned above. The 'aligned' Monte Carlo sky realizations in this paper, in addition to the Q-O alignment, did not enforce the low C_2 (as Pontzen noticed), but did effectively enforce the planar octopole via the statistics S^(3,3) in that paper. (A fair comparison would look for the covariance between all-alignment vs. all-power constrained realizations, which is what Sarkar et al did, and not some mix). So it doesn't appear that the alignments-imply-missing-correlations type of explanation is at work.

The low quadrupole is an essential ingredient to see the effect of the higher-l correlations. Otherwise the shalf value is dominated by the quadrupole.

Dragan Huterer wrote:
As for the ISW, I don't follow. In our review paper, we basically agree with Andrew's comments how low large-angle power would imply fortuitous cancellation between SW and ISW signals, and therefore "mak[ing] the prior on any theory purporting to explain it very small". But how is this an argument against the observed low correlations?

The likelihood isn't affected. The prior is. As I said at the top of this post: if you don't believe priors are important in model selection, I'm honestly not sure where to start.

As far as I can tell, in the absence of theory, it's impossible to quantify unambiguously the significance of any anomaly.

Okay, we can go back and forth about frequentist vs. Bayesian approaches to statistics until we're blue in the face, but when you're data has features that are incredibly unlikely in your best fit theory at some point you have to wake up to the fact that either your theory or your data is in serious trouble. Certainly we are much happier when we find a theory that accommodates the anomalous data, but that doesn't entitle us to say that until we have a theory the data isn't anomalous.

In this particular case, the chance of (the cut sky) Shalf being this small is a few in 10,000 in the best fit LCDM. (And to be specific, I mean the p-value.) But actually the situation is much worse. As far as we can tell (and we're double checking this.), in the vast majority of the cases where (the cut sky) Shalf manages to be this low, it is because all the low l Cl's are very low -- not at all how our (cut) sky achieves this low an Shalf. Our sky achieves it by a careful cancellation between the contributions of l=2,3,4,5 against l>=6. In other words, apparently the vast majority of simulated (cut) skies with low Shalf have pseudo-Cls that are inconsistent with the pseudo-Cls of our sky. Thus the true p-value of the lack of large angle corrleations
is << 10^{-4}.

As far as I can tell, in the absence of theory, it's impossible to quantify unambiguously the significance of any anomaly.

Okay, we can go back and forth about frequentist vs. Bayesian approaches to statistics until we're blue in the face, but when you're data has features that are incredibly unlikely in your best fit theory at some point you have to wake up to the fact that either your theory or your data is in serious trouble. Certainly we are much happier when we find a theory that accommodates the anomalous data, but that doesn't entitle us to say that until we have a theory the data isn't anomalous.

I certainly agree that frequentist statistics can be a useful guide to 'what might be expected from a Bayesian analysis if only we had a good model'. In fact, many frequentist statistics can be interpreted in exactly this way. But then why not do the actual analysis over an entire class of models to check?

Our actual Bayesian analysis suggests that physical theories won't ever 'explain' the shalf_cut value. That makes its small value much less interesting. So, by all means, call it an anomaly (otherwise we will just be disagreeing on semantics); but don't expect a theory to come along which is physically motivated, provides a better fit to the CMB as a whole, and predicts low shalf_cut.

(Of course, a theory which is non-Gaussian on large scales but highly Gaussian on small scales might come along and evade our limits. I'll eat my hat...)

In this particular case, the chance of (the cut sky) Shalf being this small is a few in 10,000 in the best fit LCDM. (And to be specific, I mean the p-value.) But actually the situation is much worse. As far as we can tell (and we're double checking this.), in the vast majority of the cases where (the cut sky) Shalf manages to be this low, it is because all the low l Cl's are very low -- not at all how our (cut) sky achieves this low an Shalf. Our sky achieves it by a careful cancellation between the contributions of l=2,3,4,5 against l>=6. In other words, apparently the vast majority of simulated (cut) skies with low Shalf have pseudo-Cls that are inconsistent with the pseudo-Cls of our sky. Thus the true p-value of the lack of large angle corrleations
is << 10^{-4}.

I can well believe this is true. If you keep staring at the data forever, you can distill the anomaly down to more and more specific aspects of our sky. The p-values should get smaller and smaller.

The trouble is, you can do that whether or not the anomaly has physical significance; you are simply making more and more specific a posteriori choices.

If you couple this sort of distillation with some sort of physical explanation of why the new statistic is more meaningful, then we are talking from the same page. If, on the other hand, you simply publish the p-values of these even-more-a-posteriori statistics, then I don't see the point.